Image (range)

Let $f:X\to Y$ be a function.

• The image (or range) of $f$ is $f(X):=\{f(x):x\in X\}\subseteq Y.$
• More generally, for a subset $A\subseteq X$, the image of $A$ under $f$ is $f(A):=\{f(a):a\in A\}\subseteq Y.$

The image captures the “actual outputs” of $f$ and is the natural codomain for which $f$ becomes surjective (if one replaces $Y$ by $f(X)$).

Examples:

• If $f:\mathbb{R}\to\mathbb{R}$, $f(x)=x^2$, then $f(\mathbb{R})=[0,\infty)$.
• If $f(x)=x^2$ and $A=[-1,2]$, then $f(A)=[0,4]$.
• If $f:\mathbb{R}\to\mathbb{R}$, $f(x)=e^x$, then $f(\mathbb{R})=(0,\infty)$.